How does the Mapping Node work?
Location does indeed just add to the vector-- using a mapping node that maps location is no different than using a vector math/add node.  Scaling just multiplies the vector's components.  Rotation of 2D coordinates moves your x,y to length * cos (thetaInitial+thetaMapping), length * sin (thetaInitial+thetaMapping).  The only thing that's remotely complicated is the order of those operations-- but then, you can always specify order by doing sequential mappings.
But really, what it's doing is creating a transformation matrix and then multiplying your vectors' positions by that matrix: https://en.wikipedia.org/wiki/Transformation_matrix .  The matrix multiplication explains the math behind 3D rotations and the order of operations.
If you're recreating the effect for a shader in, say, Unity, you can just add values to your UV (for example), there's nothing wrong with that.  But there's a lot of value in having at least some understanding of transformation matrices, for writing shaders and for understanding 3D graphics in general.